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Zbl 1156.34019
Tian, Yu; Jiang, Daqing; Ge, Weigao
Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations.
(English)
[J] Appl. Math. Comput. 200, No. 1, 123-132 (2008). ISSN 0096-3003

The authors consider the impulsive boundary value problem \aligned &-x'' + Mx = f(t,x), \quad 0 < t < 2\pi,\ t \not= t_k\\ &\triangle x\vert _{t=t_k} = I_k(x(t_k)), \quad -\triangle x'\vert _{t=t_k} = J_k(x(t_k)), \quad k = 1,2,\dots,l,\\ &x(0) = x(2\pi),\quad x'(0) = x'(2\pi), \endaligned where $0 < t_1 < \dots < t_l < 2\pi$, $M > 0$, $f: [0,2\pi] \times [0,\infty) \to [0,\infty)$, $I_k : [0,\infty) \to R$, $J_k : [0,\infty) \to [0,\infty)$ are continuous functions. Sufficient conditions for the existence of at least two positive solutions are found. The arguments are based on fixed point index theory in cones. An example illustrating the results is included.
[Jan Tomeček (Olomouc)]
MSC 2000:
*34B37 Boundary value problems with impulses
34B18 Positive solutions of nonlinear boundary value problems

Keywords: periodic boundary value problem; impulsive problem; multiple positive solutions; fixed point index

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